diff options
Diffstat (limited to 'library/topology')
| -rw-r--r-- | library/topology/basis.tex | 4 | ||||
| -rw-r--r-- | library/topology/topological-space.tex | 10 |
2 files changed, 11 insertions, 3 deletions
diff --git a/library/topology/basis.tex b/library/topology/basis.tex index 61a358f..15910f9 100644 --- a/library/topology/basis.tex +++ b/library/topology/basis.tex @@ -47,8 +47,8 @@ \end{definition} \begin{definition}\label{genopens} - $\genOpens{B}{X} = \{ U\in\pow{X} \mid \text{for all $x\in U$ there exists $V\in B$ - such that $x\in V\subseteq U$} \}$. + $\genOpens{B}{X} = \left\{ U\in\pow{X} \middle| \textbox{for all $x\in U$ there exists $V\in B$ + \\ such that $x\in V\subseteq U$}\right\}$. \end{definition} \begin{lemma}\label{emptyset_in_genopens} diff --git a/library/topology/topological-space.tex b/library/topology/topological-space.tex index e467d48..2bbdf09 100644 --- a/library/topology/topological-space.tex +++ b/library/topology/topological-space.tex @@ -11,7 +11,6 @@ such that \begin{enumerate} \item\label{opens_type} $\opens[X]$ is a family of subsets of $\carrier[X]$. - \item\label{emptyset_open} $\emptyset\in\opens[X]$. \item\label{carrier_open} $\carrier[X]\in\opens[X]$. \item\label{opens_inter} For all $A, B\in \opens[X]$ we have $A\inter B\in\opens[X]$. \item\label{opens_unions} For all $F\subseteq \opens[X]$ we have $\unions{F}\in\opens[X]$. @@ -26,6 +25,15 @@ $U$ is open in $X$ iff $U\in\opens[X]$. \end{abbreviation} +\begin{proposition}\label{emptyset_open} + Let $X$ be a topological space. + Then $\emptyset$ is open in $X$. +\end{proposition} +\begin{proof} + We have $\unions{\emptyset} = \emptyset\subseteq\opens[X]$ by \cref{unions_emptyset,emptyset_subseteq}. + Follows by \cref{opens_unions}. +\end{proof} + \begin{proposition}\label{union_open} Let $X$ be a topological space. Suppose $A$, $B$ are open. |